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Abstract

We analyze the operation of 2 × 2 switches composed of two coupled waveguides operating on the basis of parity-time (PT) symmetry: the two waveguides differ through their gain or loss factors and not through the real part of their propagation constant. Plasmonics constitutes a preferred application for such systems, since combination of plasmonics with gain is increasingly mastered. The exact PT-symmetric case (gain and loss of identical absolute value) is considered as well as various unbalanced cases, thanks to their respective switching diagrams. Although perfect signal-conserving cross and bar states are not always possible in the latter cases, they can nevertheless form the basis of very good switches if precise design rules are followed. We draw from the analysis what the optimal configurations are in terms of, e.g., guide gain or gain-length product to operate the switch. Many analytical or semi-analytical results are pointed out. A practical example based on the coupling of a long-range surface-plasmon-polariton and a polymeric waveguide having gain is provided.

Figures (9)

(a) Sketch of a PT symmetry directional coupler; (b) Real parts (“Re”, solid lines) and Imaginary parts (“Im”, dashed lines) of the two normalized eigenvalues βeff1 and βeff2 of a coupled system; (c) Same diagram presented on the whole algebraic gain axis. “GT*” refers to the gauge transform that realigns the real axis with the dotted line; (d) “Uncalibrated” case with g1/χ2 = 1.5 using the same gain axis; (e) “Uncalibrated case with g1/χ2=0.5; (f) “Biased” case with relatively small fixed losses; (g) “Biased” case with stronger fixed losses. The motion of the centre of the “Re” and “Im” patterns to the left and/or to the top is pointed out in the two last cases.

Color map of a transmission Tij(L,g1) in a dB scale. The thin black solid lines on the maps correspond to the iso-level curves Tij = 1. The ordinate axes points corresponding to a complete power crossover are marked by a Ө, and those indicating the bar state with zero net crossover are marked by a ⊗. The green and magenta color arrows correspond to a perfect switching operation from Tii = 1, Tij = 0 to Tii = 0, Tij = 1, where i≠j. Note that arrows from the two top panels (injection in the “gainy” guide 1) are identical, and differ from arrows of the two bottom ones (injection in the “lossy” guide).

Color maps of switching operation corresponding to the typical plasmonic setting with fixed loss χ2 = 0.5κ and variable gain. The thin black solid lines on the maps correspond to the iso-level curves Tij = 1. The violet arrows indicate a switching from an initial bar state Ө to a final cross state ⊗, green arrows correspond to a switching from an initial cross state ⊗ to a final bar state Ө. The lengths for the two top T1j diagrams are denoted Lp. The vertical dotted line corresponds to the “exact PT symmetry” operation point.

(a) Cross and bar states intensity variation with gain g1 for a fixed loss level χ2 = 0.42κ. (b) Cross-sectional view of the hybrid plasmonic/dielectric coupler. The modes propagate along the third dimension (the z-axis). The total length L of the device along the z axis, not shown here, is 5 mm. (c) Cross and bar states intensity vs. g = Im(εSU8). The coupler is fed by the SU8 waveguide. To compute these curves, we assumed that all the electromagnetic power contained in the left semi-infinite xy plane is carried by the SU8 waveguide while all the electromagnetic power contained in the right semi-infinite xy plane is carried by the metallic stripe.

Color maps of switching operation with fixed gain g1 = 0.8κ and variable loss (χ2 variable negative gain). The thin black solid lines on the maps correspond to the iso-level curves Tij = 1. The violet arrows indicate a switching from an initial bar state Ө to a final cross state ⊗. The vertical dotted line corresponds to the “exact PT symmetry” operation point. In the top maps, the switching shown is the only one with final unity transmission, whereas in the top map, there is not even a single such possibility.